Yes. The simplest thing to do is to light the apparatus with infrared light using something like the ANY-maze infrared illuminator. Provided your camera doesn’t have an infrared blocking filter then the video picture will look as if the animal is in normal light. A word of warning though:, some webcams do include infrared blocking filters, so you need to check that first. Alternatively use the ANY-maze USB camera which doesn’t block infrared and works very well in this type of situation.
A handy tip in this situation is to point the infrared illuminator at the ceiling and let the light reflect down onto the apparatus – this lights the apparatus more evenly than would occur if you pointed the illuminator directly at the apparatus (which tends to light the middle more than the edges – like a flashlight would).
If you don’t want to use infrared light to illuminate the apparatus then some analogue cameras are very sensitive and even in low light produce clear, well defined images. However, to use an analogue camera you do need a digitizer card.